Supercritical problems in domains with thin toroidal holes

Abstract

In this paper we study the Lane-Emden-Fowler equation $$(P{)}_{\epsilon }\{\Delta u+|u{|}^{q-2}u=0in{\mathcal{D}}_{\epsilon },u=0on\partial {\mathcal{D}}_{\epsilon }.$$ Here ${\mathcal{D}}_{\epsilon }=\mathcal{D}\setminus \{x\in \mathcal{D}:\mathrm{dist}(x,{\Gamma }_{\ell })\le \epsilon \}$, $\mathcal{D}$ is a smooth bounded domain in ${\mathbb{R}}^N$, ${\Gamma }_{\ell }$ is an $\ell -$dimensional closed manifold such that ${\Gamma }_{\ell }\subset \mathcal{D}$ with $1\le \ell \le N-3$ and $q=\frac{2(N-\ell )}{N-\ell -2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P{)}_{\epsilon }$ increases as $\epsilon$ goes to zero.

References in corpus (4)

• Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents
• Boundary clustered layers near the higher critical exponents
• Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole
• Boundary towers of layers for some supercritical problems